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Sequences

A sequenceA function whose domain is a set of consecutive natural numbers starting with 1. is a function whose domain is a set of consecutive natural numbers beginning with 1. For example, the following equation with domain {1,2,3,…} defines an infinite sequenceA sequence whose domain is the set of natural numbers {1,2,3,…}.:

a(n)=5n−3oran=5n−3

The elements in the range of this function are called terms of the sequence. It is common to define the nth term, or the general term of a sequenceAn equation that defines the nth term of a sequence commonly denoted using subscripts an., using the subscritped notation an, which reads “a sub n.” Terms can be found using substitution as follows:

The ellipsis (…) indicates that this sequence continues forever. Unlike a set, order matters. If the domain of a sequence consists of natural numbers that end, such as {1,2,3,…,k}, then it is called a finite sequenceA sequence whose domain is {1,2,3,…,k} where k is a natural number..

Example 1

Given the general term of a sequence, find the first 5 terms as well as the 100th term: an=n(n−1)2.

Solution:

To find the first 5 terms, substitute 1, 2, 3, 4, and 5 for n and then simplify.

One interesting example is the Fibonacci sequence. The first two numbers in the Fibonacci sequence are 1, and each successive term is the sum of the previous two. Therefore, the general term is expressed in terms of the previous two as follows:

Fn=Fn−2+Fn−1

Here F1=1, F2=1, and n>2. A formula that describes a sequence in terms of its previous terms is called a recurrence relationA formula that uses previous terms of a sequence to describe subsequent terms..

Example 3

Find the first 7 Fibonacci numbers.

Solution:

Given that F1=1 and F2=1, use the recurrence relation Fn=Fn−2+Fn−1 where n is an integer starting with n=3 to find the next 5 terms:

Fibonacci numbers appear in applications ranging from art to computer science and biology. The beauty of this sequence can be visualized by constructing a Fibonacci spiral. Consider a tiling of squares where each side has a length that matches each Fibonacci number:

Connecting the opposite corners of the squares with an arc produces a special spiral shape.

This shape is called the Fibonacci spiral and approximates many spiral shapes found in nature.

Series

A seriesThe sum of the terms of a sequence. is the sum of the terms of a sequence. The sum of the terms of an infinite sequence results in an infinite seriesThe sum of the terms of an infinite sequence denoted S∞., denoted S∞. The sum of the first n terms in a sequence is called a partial sumThe sum of the first n terms in a sequence denoted Sn., denoted Sn. For example, given the sequence of positive odd integers 1, 3, 5,… we can write:

S∞=1+3+5+7+9+⋯InfiniteseriesS5=1+3+5+7+9=255thpartialsum

Example 4

Determine the 3rd and 5th partial sums of the sequence: 3,−6, 12,−24, 48,…

Solution:

S3=3+(−6)+12=9S5=3+(−6)+12+(−24)+48=33

Answer: S3=9; S5=33

If the general term is known, then we can express a series using sigmaA sum denoted using the symbol Σ (upper case Greek letter sigma). (or summationUsed when referring to sigma notation.) notation:

S∞=Σn=1∞n2=12+22+32+…InfiniteseriesS3=Σn=13n2=12+22+323rdpartialsum

The symbol Σ (upper case Greek letter sigma) is used to indicate a series. The expressions above and below indicate the range of the index of summationThe variable used in sigma notation to indicate the lower and upper bounds of the summation., in this case represented by n. The lower number indicates the starting integer and the upper value indicates the ending integer. The nth partial sum Sn can be expressed using sigma notation as follows:

Sn=Σk=1nak=a1+a2+⋯+an

This is read, “the sum of ak as k goes from 1 to n.” Replace n with ∞ to indicate an infinite sum.

Example 8

Key Takeaways

A sequence is a function whose domain consists of a set of natural numbers beginning with 1. In addition, a sequence can be thought of as an ordered list.

Formulas are often used to describe the nth term, or general term, of a sequence using the subscripted notation an.

A series is the sum of the terms in a sequence. The sum of the first n terms is called the nth partial sum and is denoted Sn.

Use sigma notation to denote summations in a compact manner. The nth partial sum, using sigma notation, can be written Sn=Σk=1nak. The symbol Σ denotes a summation where the expression below indicates that the index k starts at 1 and iterates through the natural numbers ending with the value n above.

Topic Exercises

Part A: Sequences

Find the first 5 terms of the sequence as well as the 30th term.

an=2n

an=2n+1

an=n2−12

an=n2n−1

an=(−1)n(n+1)2

an=(−1)n+1n2

an=3n−1

an=2n−2

an=(12)n

an=(−13)n

an=(−1)n−13n−1

an=2(−1)nn+5

an=1+1n

an=n2+1n

Find the first 5 terms of the sequence.

an=2x2n−1

an=(2x)n−1

an=xnn+4

an=x2nx−2

an=nx2nn+1

an=(n+1)xnn2

an=(−1)nx3n

an=(−1)n−1xn+1

Find the first 5 terms of the sequence defined by the given recurrence relation.

an=an−1+5 where a1=3

an=an−1−3 where a1=4

an=3an−1 where a1=−2

an=−2an−1 where a1=−1

an=nan−1 where a1=1

an=(n−1)an−1 where a1=1

an=2an−1−1 where a1=0

an=3an−1+1 where a1=−1

an=an−2+2an−1 where a1=−1 and a2=0

an=3an−1−an−2 where a1=0 and a2=2

an=an−1−an−2 where a1=1 and a2=3

an=an−2+an−1+2 where a1=−4 and a2=−1

Find the indicated term.

an=2−7n; a12

an=3n−8; a20

an=−4(5)n−4; a7

an=6(13)n−6; a9

an=1+1n; a10

an=(n+1)5n−3; a5

an=(−1)n22n−3; a4

an=n(n−1)(n−2); a6

An investment of $4,500 is made in an account earning 2% interest compounded quarterly. The balance in the account after n quarters is given by an=4500(1+0.024)n. Find the amount in the account after each quarter for the first two years. Round to the nearest cent.

The value of a new car after n years is given by the formula an=18,000(34)n. Find and interpret a7. Round to the nearest whole dollar.

The number of comparisons a computer algorithm makes to sort n names in a list is given by the formula an=nlog2n. Determine the number of comparisons it takes this algorithm to sort 2×106 (2 million) names.

The number of comparisons a computer algorithm makes to search n names in a list is given by the formula an=n2. Determine the number of comparisons it takes this algorithm to search 2×106 (2 million) names.

Part B: Series

Find the indicated partial sum.

3, 5, 9, 17, 33,…; S4

−5, 7, −29, 79, −245,…; S4

4, 1, −4, −11, −20,…; S5

0, 2, 6, 12, 20,…; S3

an=2−7n; S5

an=3n−8; S5

an=−4(5)n−4; S3

an=6(13)n−6; S3

an=1+1n; S4

an=(n+1)5n−3; S3

an=(−1)n22n−3; S5

an=n(n−1)(n−2); S4

Evaluate.

∑k=153k

∑k=162k

∑i=26i2

∑i=04(i+1)2

∑n=15(−1)n+12n

∑n=510(−1)nn2

∑k=−22(12)k

∑k=−40(13)k

∑k=04(−2)k+1

∑k=−13(−3)k−1

∑n=153

∑n=17−5

∑k=−23k(k+1)

∑k=−22(k−2)(k+2)

Write in expanded form.

∑n=1∞n−1n

∑n=1∞n2n−1

∑n=1∞(−12)n−1

∑n=0∞(−23)n+1

∑n=1∞3(15)n

∑n=0∞2(13)n

∑k=0∞(−1)kxk+1

∑k=1∞(−1)k+1xk−1

∑i=0∞(−2)i+1xi

∑i=1∞(−3)i−1x3i

∑k=1∞(2k−1)x2k

∑k=1∞kxk−1k+1

Express the following series using sigma notation.

x+2x2+3x3+4x4+5x5

12x2+23x3+34x4+45x5+56x6

2+22x+23x2+24x3+25x4

3x+32x2+33x3+34x4+35x5

2x+4x2+8x3+⋯+2nxn

x+3x2+9x3+⋯+3nxn+1

5+(5+d)+(5+2d)+⋯+(5+nd)

2+2r1+2r2+⋯+2rn−1

34+38+316+⋯+3(12)n

83+164+325+⋯+2nn

A structured settlement yields an amount in dollars each year, represented by n, according to the formula pn=10,000(0.70)n−1. What is the total amount gained from the settlement after 5 years?

The first row of seating in a small theater consists of 14 seats. Each row thereafter consists of 2 more seats than the previous row. If there are 7 rows, how many total seats are in the theater?

Part C: Discussion Board

Research and discuss Fibonacci numbers as they are found in nature.

Research and discuss the life and contributions of Leonardo Fibonacci.

Explain the difference between a sequence and a series. Provide an example of each.